The 6.8 magnitude quake, centred on the town of Sivrice in Elazig province, caused buildings to collapse and sent residents rushing into the street.
Tremors were also felt in neighbouring Syria, Lebanon and Iran.
Earthquakes are common in Turkey – about 17,000 people died in a massive quake in the western city of Izmit in 1999.
Friday’s quake struck at about 20:55 local time (17:55 GMT).
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According to Turkey’s Disaster and Emergency Management Authority (Afad), there were more than 200 aftershocks recorded after the earthquake.
It said more than 400 rescue teams are heading towards the region, along with beds and tents for those displaced. It warned people not to return to damaged buildings in case of aftershocks.
Afad said that 17 people had died in Elazig province, while the official count in Malatya is four. The governor or Malatya, however, earlier said that six people had died there.
The region struck by the quake, some 550km (340 miles) east of the capital Ankara, is remote and sparsely populated, so details of damage and fatalities could be slow to emerge.
Officials have sent beds, tents and blankets to the area, where overnight temperatures regularly fall below zero. In its advisory on Saturday morning, the emergency authority said the temperature had fallen to -8C.
The Turkish Red Crescent has also dispatched hundreds of personnel with emergency supplies, it said.
Sivrice, a town of about 4,000 people, is a popular tourist spot on the shore of Hazar lake, the source of the river Tigris.